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A385430 Least number k such that k and k + n! have the same number of divisors.

%I A385430 #29 Aug 23 2025 23:11:36
%S A385430 2,3,5,5,7,7,11,23,17,11,17,29,46,19,43,23,31,37,89,29,31,31,97,62,41,
%T A385430 59,47,67,159,107,127,79,37,97,61,131,86,43,97,53,61,97,71,47,94,101,
%U A385430 233,53,83,61,249,53,71,158,71,149,107,134,254,206,166,131,271
%N A385430 Least number k such that k and k + n! have the same number of divisors.
%C A385430 Inspired by A284783.
%C A385430 First differs from A037153 at n=13 (and when they differ a(n) is a composite < A037153(n)).
%H A385430 Sean A. Irvine, <a href="/A385430/b385430.txt">Table of n, a(n) for n = 1..82</a>
%e A385430 a(1) = 2 since d(2) = d(3) = 2;
%e A385430 a(5) = 7 since d(7) = d(7+5!) = 2;
%e A385430 a(13) = 46 since d(46) = d(46+13!) = 4; etc.
%t A385430 a[n_] := Block[{k = 2}, While[ DivisorSigma[0, k] != DivisorSigma[0, k + n!], k++]; k]; Array[ a, 51]
%o A385430 (PARI) a(n) = my(k=1); while (numdiv(k) != numdiv(k+n!), k++); k; \\ _Michel Marcus_, Aug 02 2025
%Y A385430 Cf. A000005, A000142, A006881, A037153, A284783.
%K A385430 nonn,changed
%O A385430 1,1
%A A385430 _Robert G. Wilson v_, Jul 31 2025
%E A385430 More terms from _Sean A. Irvine_, Aug 08 2025